Given the function $f (x) = - \frac{4}{x^{2}}$ $f(x) = - 4 / x^2$ , how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?

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Answer 1 · 2016-11-11T20:44:43

See below.

$f$ $f$ is continuous at every real number except $0$ $0$ .
$0$ $0$ is not in $[1, 4]$ $[1,4]$ .
Therefore, $f$ $f$ is continuous on $[1, 4]$ $[1,4]$

$f'(x) = 8x^-3$ is defined for all $x$ other than $0$ .
$0$ is not in $[1,4]$ .
Therefore, $f$ is differentiable on $(1,4)$

To find $c$ , solve $f'(x) = (f(4)-f(1))/(4-1)$ . The solutions in $(1,4)$ are values of $c$ .

Solving, we get $x=root(3)(32/5)$

Given the function f(x) = - 4 / x^2f(x)=−4x2f(x) = - 4 / x^2, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?